"Seems to me that a cross bike with 700x36 tyres wuld go quite well"
Why? Why buy and setup a cross bike as a touring bike? Why not buy a touring bike and use it as a touring bike?
If a bike is a true cross bike, it will have a high bottom bracket with normal cross tires such as 30mm. With even larger tires, it will have a really high bottom bracket. Maybe not a huge concern, but most dedicated touring bikes have lower bottom brackets for stability even with large tires.
If you are talking about getting a so called cross bike that really is just a road bike of sorts with cantilever brake mounts, then there is problems with that too. Cross bikes come with 130mm rear spacing. Road bike spacing. You can remove spacers from 135mm mountain bike rear hubs to get it to fit hopefully. But some 135mm rear hubs you cannot remove spacers. If you know you want the stronger, wider dished rear wheels available with 135mm spacing, why get a cross bike with 130mm spacing?
Are you thinking you can have just one bike to do it all and change tires and have the best of all worlds? It does not work that way. If a bike is setup for loaded touring, its not as fun to ride as a sporty/racy bike. If 99% of your miles are fun recreational riding, then get a bike that fits that type of riding. Worry about that other 1% when the time comes. I'm not a fan of having one bike supposedly doing everything. Jack of all trades, master of none.
Don't buy a bike for a tour that may never occur. Get one for the riding you are doing now. Enjoy your riding now.
Now, will a cross bike work as a touring bike? Sure. Almost any bike will work as a touring bike. If you already own a cross bike, use it to tour. Or ride around the town or to work, etc. No need to buy new bikes unless you want a new bike.